About the Pricing Equations in Finance

In this article we study a decoupled forward backward stochastic differential equation (FBSDE) and the associated

system of partial integro-differential obstacle problems

, in a flexible Markovian set-up made of a jump-diffusion with regimes. These equations are motivated by numerous applications in financial modeling, whence the title of the paper. This financial motivation is developed in the first part of the paper, which provides a synthetic view of the theory of pricing and hedging financial derivatives, using backward stochastic differential equations (BSDEs) as main tool. In the second part of the paper, we establish the well-posedness of reflected BSDEs with jumps coming out of the pricing and hedging problems exposed in the first part. We first provide a construction of a Markovian model made of a jump-diffusion – like component

X

interacting with a continuous-time Markov chain – like component

N

. The jump process

N

defines the so-called

regime

of the coefficients of

X

, whence the name of

jump-diffusion with regimes

for this model. Motivated by

optimal stopping

and

optimal stopping game

problems (pricing equations of

American or game contingent claims

), we introduce the related

reflected and doubly reflected Markovian BSDEs

, showing that they are

well-posed

in the sense that they have

unique solutions, which depend continuously on their input data.

As an aside, we establish the

Markov property

of the model. In the third part of the paper we derive the related

variational inequality approach

. We first introduce the systems of partial integro-differential variational inequalities formally associated to the reflected BSDEs, and we state suitable definitions of viscosity solutions for these problems, accounting for jumps and/or systems of equations. We then show that the state-processes (first components

Y

) of the solutions to the reflected BSDEs can be characterized in terms of the

value functions

of related optimal stopping or game problems, given as

viscosity solutions with polynomial growth

to related integro-differential obstacle problems. We further establish a

comparison principle

for semi-continuous viscosity solutions to these problems, which implies in particular the

uniqueness

of the viscosity solutions. This comparison principle is subsequently used for proving the convergence of

stable, monotone and consistent

approximation schemes to the value functions. Finally in the last part of the paper we provide various extensions of the results needed for applications in finance to pricing problems involving

discrete dividends

on a financial derivative or on the underlying asset, as well as various forms of

discrete path-dependence

.